A priori Estimates for the Incompressible Free-Boundary Magnetohydrodynamics Equations with Surface Tension

TL;DR

This paper establishes a priori estimates for 3D incompressible free-boundary MHD equations with surface tension, reducing regularity requirements and providing the first such results for ideal free-boundary MHD.

Contribution

It introduces novel a priori estimates for free-boundary MHD with surface tension, eliminating the need for extra regularity on the flow map.

Findings

First a priori estimates for free-boundary MHD with surface tension

Reduction of regularity requirements for the flow map

Establishment of estimates in Lagrangian coordinates with H^{3.5} regularity

Abstract

We consider the three-dimensional incompressible free-boundary magnetohydrodynamics (MHD) equations in a bounded domain with surface tension on the boundary. We establish a priori estimate for solutions in the Lagrangian coordinates with $H^{3.5}$ regularity. To the best of our knowledge, this is the first result focusing on the incompressible ideal free-boundary MHD equations with surface tension. It is worth pointing out that the $1/2$-extra spatial regularity for the flow map $\eta$ is no longer required in this manuscript thanks to the presence of the surface tension on the boundary.

Equations

$$\{\begin{array}{ll}{\partial }_{t}u+u\cdot \nabla u-B\cdot \nabla B+\nabla (p+\frac{1}{2}|B{|}^2)=0& \text{in}\mathcal{D};\\ {\partial }_{t}B+u\cdot \nabla B-B\cdot \nabla u=0,& \text{in}\mathcal{D};\\ \text{div }u=0,\text{div }B=0,& \text{in}\mathcal{D},\end{array}$$

$$\Omega (0)=\{x:(0,x)\in \mathcal{D}\},(u,B)=(u_0,B_0),\text{in}\{0\}\times {\Omega }_0.$$

$$\{\begin{array}{ll}({\partial }_t+u\cdot \nabla ){|}_{\partial \mathcal{D}}\in \mathcal{T}(\partial \mathcal{D})& \\ P=\sigma \mathcal{H}& \text{on}\partial \mathcal{D},\\ B\cdot \mathcal{N}=0& \text{on}\partial \mathcal{D},\end{array}$$

$$\frac{d}{dt}\left(\frac{1}{2}{\int }_{\Omega (t)}|u{|}^2+\frac{1}{2}{\int }_{\Omega (t)}|B{|}^2+\sigma {\int }_{\partial \Omega (t)}dS(\Omega (t))\right)$$

$$={\int }_{\Omega (t)}u\cdot D_{t}u+{\int }_{\Omega (t)}B\cdot D_{t}B+\frac{d}{dt}\left(\sigma {\int }_{\partial \Omega (t)}dS(\Omega (t))\right)$$

$$=-{\int }_{\Omega (t)}u\cdot \nabla (p+\frac{1}{2}|B{|}^2)+{\int }_{\Omega (t)}u\cdot (B\cdot \nabla B)+{\int }_{\Omega (t)}B\cdot (B\cdot \nabla u)$$

$$+\frac{d}{dt}\left(\sigma {\int }_{\partial \Omega (t)}dS(\Omega (t))\right)$$

$$=-{\int }_{\partial \Omega (t)}(u\cdot \mathcal{N})P+\frac{d}{dt}\left(\sigma {\int }_{\partial \Omega (t)}dS(\Omega (t))\right)$$

$$+{\int }_{\Omega (t)}u\cdot (B\cdot \nabla B)-{\int }_{\Omega (t)}u\cdot (B\cdot \nabla B)=0,$$

$$-{\nabla }_{\mathcal{N}}(p+\frac{1}{2}|B{|}^2)\ge {ϵ}_0>0,$$

$${\partial }_t\eta (t,y)=u(t,\eta (t,y)),\eta (0,y)=y.$$

$$v(t,y)=u(t,\eta (t,y)),b(t,y)=B(t,\eta (t,y)),q(t,y)=p(t,\eta (t,y)).$$

$${\partial }_{\mu }{a}^{\mu \alpha }=0.$$

$$\{\begin{array}{ll}{\partial }_t{v}_{\alpha }-b_{\beta }{a}^{\mu \beta }{\partial }_{\mu }{b}_{\alpha }+a_{\alpha }^{\mu }{\partial }_{\mu }Q=0& \text{in}[0,T]\times \Omega ;\\ {\partial }_t{b}_{\alpha }-b_{\beta }{a}^{\mu \beta }{\partial }_{\mu }{v}_{\alpha }=0& \text{in}[0,T]\times \Omega ;\\ a^{\mu \alpha }{\partial }_{\mu }{v}_{\alpha }=0,a^{\mu \alpha }{\partial }_{\mu }{b}_{\alpha }=0& \text{in}[0,T]\times \Omega ;\\ v_3=0,b_3=0& \text{on}{\Gamma }_0;\\ a^{\mu \alpha }{N}_{\mu }Q+\sigma (\sqrt{g}{\Delta }_g{\eta }^{\alpha })=0& \text{on}\Gamma ;\\ a^{\mu \nu }{b}_{\nu }{N}_{\mu }=0& \text{on}\Gamma ,\end{array}$$

$$g_{ij}={\partial }_i{\eta }^{\mu }{\partial }_j{\eta }_{\mu },{\Delta }_g(\cdot )=\frac{1}{\sqrt{g}}{\partial }_i(\sqrt{g}{g}^{ij}{\partial }_j(\cdot )),\text{ where }g:=\det (g_{ij}).$$

$$\Omega ={\mathbb{T}}^2\times (0,1),$$

$$N(t)=\Vert \eta {\Vert }_{3.5}^2$$

$$+\Vert Q{\Vert }_{3.5}^2+\Vert Q_t{\Vert }_{2.5}^2+\Vert Q_{tt}{\Vert }_1^2.$$

Full text

1. 1 Introduction

2. 1.1 History and background

3. 1.2 MHD system in Lagrangian coordinates and the main result

4. 1.3 Strategy and organisation of the paper

1 Introduction

The goal of this manuscript is to investigate the solutions in Sobolev spaces for the following incompressible inviscid MHD equations in a moving domain with surface tension on the boundary:

[TABLE]

describing the motion of conducting fluids in an electromagnetic field, where $\mathcal{D}={\cup}_{0\leq t\leq T}\{t\}\times\Omega(t)$ and $\Omega(t)\subset\mathbb{R}^{3}$ is the domain occupied by the fluid whose boundary $\partial\Omega(t)$ moves with the velocity of the fluid. Under this setting, the fluid velocity $u=(u_{1},u_{2},u_{3})$, the magnetic field $B=(B_{1},B_{2},B_{3})$, the fluid pressure $p$ and the domain $\mathcal{D}$ are to be determined; in other words, given a simply connected bounded domain $\Omega(0)\subset\mathbb{R}^{3}$ and the initial data $u_{0}$ and $B_{0}$ satisfying the constraints $\text{div}\,u_{0}=0$ and $\text{div}\,B_{0}=0$, we want to find a set $\mathcal{D}$ and the vector fields $u$ and $B$ solving (1.1) satisfying the initial conditions:

[TABLE]

The quantity $P:=p+\frac{1}{2}|B|^{2}$ (i.e., the total pressure) plays an important role here in our analysis. It determines the acceleration of the moving surface boundary.

We also require the following boundary conditions on the free boundary $\partial\mathcal{D}={\cup}_{0\leq t\leq T}\{t\}\times\partial\Omega(t)$:

[TABLE]

where $\mathcal{N}$ is the exterior unit normal to $\partial\Omega(t)$, $\sigma>0$ is the coefficient of surface tension, $\mathcal{H}$ is the mean curvature of the moving boundary embedded in $\mathbb{R}^{3}$. The first condition of (1.3) means that the boundary moves with the velocity of the fluid. The second condition in (1.3) suggests that the motion of the fluid is under the influence of the surface tension, as opposed to the case without surface tension ($\sigma=0$). Also, we remark here that $\mathcal{H}$ is a function of the unknowns and thus not known a priori. The third condition of (1.3), $B\cdot\mathcal{N}=0$ on $\partial\Omega(t)$ implies that the fluid is a perfect conductor; in other words, the induced electric field $\mathcal{E}$ satisfies $\mathcal{E}\times\mathcal{N}=0$ on $\partial\Omega(t)$.

The physical energy is conserved, i.e., denoting $D_{t}:=\partial_{t}+u\cdot\nabla$, and invoking the divergence free condition for both $u$ and $B$, we have:

[TABLE]

where the first line in the last equality vanishes as shown in (2.1) in the paper [shatah2008geometry].

This motivates the construction of the higher order energy for (1.1)-(1.3). We refer Section LABEL:tang for the details.

1.1 History and background

The MHD equations describe the behavior of an electrically conducting fluid (e.g., a plasma) acted on by a magnetic field. In particular, the free-boundary MHD equations (also known as the plasma-interface problem) describe the phenomenon when the conducting fluid is separated from the outside wall by a vacuum.

An overview of the previous results

In the absence of the magnetic field, i.e. $B=0$ in (1.1), the problem reduces to the well-known incompressible free-boundary Euler equations, whose local well-posedness in Sobolev spaces was obtained first by Wu [wu1997LWPww, wu1999LWPww] for the irrotational case with $\sigma=0$, assuming the physical sign condition $-\nabla_{N}p\geq\epsilon_{0}>0$ holds on $\partial\mathcal{D}_{t}$. This condition plays a crucial role in establishing the above well-posedness results for Euler equations. It was found by Ebin [ebin1987equations] that the incompressible Euler equations is ill-posed when physical sign condition fails. Extensions including the case without the irrotationalility assumption have been studied extensively in the past two decades, without attempting to be exhaustive, we refer [christodoulou2000motion, coutand2007LWP, kukavica2017local, lindblad2002, lindblad2005well, zhang2008free] for more details.

On the other hand, the free boundary Euler equations behaves differently when $\sigma>0$. The surface tension is known to have regularizing effect on the moving surface. As a consequence, the physical sign condition is no longer needed when establishing the local well-posedness. We refer [SchweizerFreeEuler, shatah2008geometry, shatah2008priori, shatah2011local] for more details.

The free-boundary MHD equations, nevertheless, is far less well-understood than the free-boundary Euler equations. When $\sigma=0$, under the physical sign condition111In [hao2017motion, sun2017well], the authors studied the a priori energy estimate and local well-posedness, respectively, for the free-boundary MHD equations with nontrivial vacuum magnetic field under different stabilising assumptions.

[TABLE]

Hao-Luo [hao2014priori] proved the a priori energy estimate with $H^{4}$ initial data and the local-wellposedness was established by Secchi-Trakhinin [secchi2013well] and Gu-Wang [gu2016construction]. Also, we remark here that in [hao2018ill], the authors proved that this problem is ill-posedness when (1.4) is violated in the case of dimension 2. We also mention here that in [luozhang2019MHD2.5], we proved a priori estimate with minimal regularity assumptions on the initial data (i.e., $v_{0},B_{0}\in H^{2.5+\delta}$) in a small fluid domain. Unlike the Euler equations, this assumption on the smallness of the volume of the fluid is crucial here due to the physical sign condition is unable to stabilize the MHD flow under low regularity assumptions. We will discuss more about this in the following paragraphs.

In [GuoMHDSTviscous], the authors proved the global well-posedness and exponentially decaying rate of the viscous and resistive free-boundary MHD equations when $\sigma>0$. In that paper, the kinematic viscosity and the magnetic diffusion allow them to control the enhanced regularity of the flow map, while this is impossible in the case of inviscid MHD without magnetic diffusion. In [chendingMHDST], Chen-Ding studied the vanishing viscosity-resistivity limit and the convergence rate for the viscous, resistive, free-boundary MHD system with surface tension in a half-space domain. Wang-Xin [wangxinMHDST1] studied the global well-posedness of the incompressible resistive MHD system with surface tension near the equilibrium solution (constant strength magnetic field). However, to the best of our knowledge, NO result that concerns the well-posedness theory of inviscid and non-resistive free-boundary MHD equations when $\sigma>0$ is available.

Our results

The goal of this manuscript is to establish a priori energy estimates for (1.1)-(1.3) with fixed $\sigma>0$ when $u_{0},B_{0}\in H^{3.5}(\Omega(0))$. Our result is an important first step to prove the local well-posedness for free-boundary MHD equations with surface tension, since the real conducting fluids have surface tension while the case without surface tension is just an idealized model. Moreover, we will show that the surface tension, in fact, has a stronger regularizing effect compare to that provided by the physical sign condition (1.4): We are able to get a better control of the normal component of the velocity field on the moving boundary through the boundary elliptic estimate due to the appearance of surface tension. As a result, our energy constructed in Chapter LABEL:tang contains at least two time derivatives, which removes the requirement of the flow map has to be $1/2$-derivatives more regular than $v$, $b$ in the case of no surface tension. We will give more illustration on this in Section 1.3.

1.2 MHD system in Lagrangian coordinates and the main result

We reformulate the MHD equations in Lagrangian coordinates, in which the free domain becomes fixed. Let $\Omega$ be a bounded domain in $\mathbb{R}^{3}$. Denoting coordinates on $\Omega$ by $y=(y_{1},y_{2},y_{3})$, we define $\eta:[0,T]\times\Omega\to\mathcal{D}$ to be the flow map of the velocity $u$, i.e.,

[TABLE]

We introduce the Lagrangian velocity, magnetic field and fluid pressure, respectively, by

[TABLE]

Let $\partial$ be the spatial derivative with respect to $y$ variable. We introduce the cofactor matrix $a=[\partial\eta]^{-1}$, which is well-defined since $\eta(t,\cdot)$ is almost the identity map when $t$ is sufficiently small. It’s worth noting that $a$ verifies the Piola’s identity, i.e.,

[TABLE]

Here, the Einstein summation convention is used for repeated upper and lower indices. In above and throughout, all Greek indices range over 1, 2, 3, and the Latin indices range over 1, 2.

Denote the total pressure $P=p+\frac{1}{2}|B|^{2}$ and let $Q=P(t,\eta(t,y))$. Then (1.1)-(1.3) can be reformulated as:

[TABLE]

where $N$ is the unit outer normal vector to $\partial\Omega$, $a^{T}$ is the transpose of $a$, $|\cdot|$ is the Euclidean norm and $\Delta_{g}$ is the Laplacian of the metric $g_{ij}$ induced on $\partial\Omega(t)$ by the embedding $\eta$. Specifically, we have:

[TABLE]

For the details to derive the fifth equation of (1.8) (the surface tension equation), we refer to Lemma 2.5 in [disconzi2017prioriI] for readers.

For the sake of simplicity and clean notation, here we consider the model case when

[TABLE]

where $\partial\Omega=\Gamma_{0}\cup\Gamma$ and $\Gamma=\mathbb{T}^{2}\times\{1\}$ is the top (moving) boundary, $\Gamma_{0}=\mathbb{T}^{2}\times\{0\}$ is the fixed bottom. Using a partition of unity, e.g., [DKT], a general domain can also be treated with the same tools we shall present. However, choosing $\Omega$ as above allows us to focus on the real issues of the problem without being distracted by the cumbersomeness of the partition of unity. Let $N$ stands for the outward unit normal of $\partial\Omega$. In particular, we have $N=(0,0,-1)$ on $\Gamma_{0}$ and $N=(0,0,1)$ on $\Gamma$.

In this paper, we prove:

Theorem 1.1**.**

Let $\Omega$ be defined as in (1.10). Assume that $v_{0}\in H^{3.5}(\Omega)\cap H^{4}(\Gamma)$ and $b_{0}\in H^{3.5}(\Omega)$ be divergence free vector fields with $b_{0}\cdot N=0$ on $\partial\Omega$. Assume $(\eta,v,b,Q)$ to be any solution of (1.8) with initial data $v_{0}$ and $b_{0}$. Define

[TABLE]

Then there exists a $T>0$, chosen sufficiently small, such that $N(t)\leq C_{0}$ for all $t\in[0,T]$, where $C_{0}$ only depends on $\|v_{0}\|_{3.5},\|b_{0}\|_{3.5},\|v_{0}\|_{4,\Gamma}$. Here we denote $\|f\|_{s}:=\|f(t,\cdot)\|_{H^{s}(\Omega)}$ for any function $f(t,y)\text{ on }[0,T]\times\Omega$, and $\|f\|_{s,\Gamma}:=\|f(t,\cdot)\|_{H^{s}(\Gamma)}$ for any $f(t,y)\text{ on }[0,T]\times\Gamma$.

1.3 Strategy and organisation of the paper

Notations.

All definitions and notations will be defined as they are introduced. In addition, a list of symbols will be given at the end of this section for a quick reference.

Definition 1.1**.**

The $L^{2}$- based Sobolev spaces are denoted by $H^{s}(\Omega)$, where we abbreviate corresponding norm $\|\cdot\|_{H^{r}(\Omega)}$ as $\|\cdot\|_{r}$ when no confusion can arise. We denote by $H^{s}(\Gamma)$ the Sobolev space of functions defined on the boundary $\Gamma$ ($\Gamma$=$\Gamma_{0}$ or $\Gamma$), with norm $\|\cdot\|_{s,\Gamma}$.